Two parallel plates are 4.85 cm apart. The bottom plate is charged positively and the top plate is charged negatively, producing a uniform electric field of 5.51 x 104 N/C in the region between the plates. What is the time required for an electron, which starts at rest at the upper plate, to reach the lower plate? (Assume a vacuum exists between the plates.)
To find the time required for an electron to reach the lower plate, we need to determine the distance it needs to travel and its acceleration.
First, let's determine the distance between the plates. Given that the plates are parallel and 4.85 cm apart, the distance is 0.0485 m.
Next, we need to find the acceleration of the electron. We know that the electric field between the plates is 5.51 x 10^4 N/C. The electric field is defined as the force per unit charge, so we can use it to find the force experienced by the electron.
The force experienced by the electron can be calculated using the equation F = qE, where F is the force, q is the charge, and E is the electric field. The charge of an electron is -1.6 x 10^-19 C (coulombs), so we can substitute these values into the equation:
F = (-1.6 x 10^-19 C) * (5.51 x 10^4 N/C)
F = - 8.816 x 10^-15 N
We know from Newton's second law (F = ma) that force is equal to mass multiplied by acceleration. The mass of an electron is approximately 9.11 x 10^-31 kg. So we can substitute these values into the equation:
-8.816 x 10^-15 N = (9.11 x 10^-31 kg) * a
Solving for the acceleration (a):
a = -8.816 x 10^-15 N / 9.11 x 10^-31 kg
a ≈ - 9.6723 x 10^15 m/s^2
The acceleration is negative because the force experienced by the electron is opposite to the direction of motion.
Now, let's use the kinematic equation to find the time required for the electron to travel the distance between the plates. The kinematic equation we'll use is:
v^2 = u^2 + 2as
where:
u = initial velocity (0 m/s, as the electron starts at rest)
v = final velocity
a = acceleration
s = distance
Simplifying the equation to solve for time (t):
v^2 = 0 + 2as
v^2 = 2as
v = √(2as)
Substituting the values we know:
v = √(2 * (-9.6723 x 10^15 m/s^2) * 0.0485 m)
v = √(-9.3767 x 10^14 m^2/s^2)
The velocity is imaginary (square root of a negative number) because the acceleration and distance are in opposite directions. This indicates that the electron will never reach the lower plate. Instead, it will oscillate back and forth between the plates, unable to escape the electric field.